The parametric equations for the line are:
x(t) = 5 + t, y(t) = 5 - t and z(t) = t
Given that the line through (5, 5, 0) and perpendicular to both i + j and j + k
We know that the symmetric form of the equation of the line is given by,
[tex]\frac{x-x_1}{a} =\frac{y-y_1}{b} =\frac{z-z_1}{c} =t[/tex]
Let us assume that x₁ = 5, y₁ = 5 and z₁ = 0
Evaluating the product of ( i + j) and ( j + k ) we get,
( i + j) × ( j + k )
= i - j + k
The value of a = 1, b = -1 and c = 1.
Substitute all the values in the above equation we get,
[tex]\frac{x-x_1}{a} =\frac{y-y_1}{b} =\frac{z-z_1}{c} =t[/tex]
[tex]\frac{x-5}{1} =\frac{y-5}{-1} =\frac{z-0}{1} =t[/tex]
so, the parametric form of the equation would be,
x(t) = x₁ + at
x(t) = 5 + (1)t
x(t) = 5 + t
y(t) = y₁ + bt
y(t) = 5 + (-1)t
y(t) = 5 - t
And z(t) = z₁ + ct
z(t) = 0 + (1)t
z(t) = t
Thus, the parametric equations: x(t) = 5 + t, y(t) = 5 - t and z(t) = t
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The complete question is:
Find parametric equations and symmetric equations for the line. (Use the parameter t.) The line through (5, 5, 0) and perpendicular to both i + j and j + k
x(t), y(t), z(t) = ?
A bicycle shop keeps the same ratio of wheels to bikes in stock. The ratio of wheels to bikes is always 4:1. How many wheels does the shop have if there are 18 bikes in stock?
Is it
4.5
42
72
76
Answer:
c (72)
Step-by-step explanation:
since there are four wheels to every bike, you just have to multiply the number of bikes to the amount of wheels per bike. given this, 18 * 4=72.
In how many months the amount of Rs1000 at the rate of 10%will be RS1200
Answer: To determine how many months it will take for an initial investment of Rs1000 to grow to Rs1200 at a 10% interest rate, we can use the formula:
Time = (ln(final value / initial value)) / (ln(1 + interest rate))
where ln is the natural logarithm function.
Plugging in the given values:
Time = (ln(1200/1000)) / (ln(1 + 0.10))
Time = (ln(1.2)) / (ln(1.10))
Time = 0.18232 / 0.09531
Time = 1.913 months
It will take approximately 1.913 months for an investment of Rs1000 to grow to Rs1200 at a 10% interest rate.
Step-by-step explanation:
Find the domain of y considered as a function over the reals. (Enter your answer using interval notation.)
A solution of the first order IVP consisting of this differential equation is y = [tex]\frac{1}{x^2-5}[/tex] and the domain of y considered as a function over the reals is (-∞, -√5) ∪ (-√5, +√5) ∪ (√5, ∞).
We have to find the domain of y considered as a function over the reals.
The differential equation is y' + 2x[tex]y^2[/tex] = 0
The solution of the given differential equation is y = 1/([tex]x^2[/tex] + c)......(1)
The given initial value; y(-3) = 1/4
Replace the initial value in the first-order DE solution provided. To obtain the equation for y, locate the constant c and then re-substitute 'c'.
Now put x = -3 and y(-3) = 1/4 in equation 1
[tex]\frac{1}{4}=\frac{1}{(-3)^2+c}[/tex]
[tex]\frac{1}{4}=\frac{1}{9+c}[/tex]
9+c = 4
Subtract 9 on both side, we get
c = -5
Now substitute the value of c again in equation 1
y = [tex]\frac{1}{x^2-5}[/tex]
To find the domain of y we can write it as
y = [tex]\frac{1}{(x)^2-(\sqrt{5})^2}[/tex]
Now the simplified form is
y = [tex]\frac{1}{(x-\sqrt{5})(x+\sqrt{5})}[/tex]
Now the domain of y over R is
(-∞, -√5) ∪ (-√5, +√5) ∪ (√5, ∞).
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The complete question is:
y = 1/([tex]x^2[/tex] + c) is a one parameter family of solutions of the first order DE y' + 2x[tex]y^2[/tex] = 0. Find a solution of the first order IVP consisting of this differential equation and the given initial condition.
y(-3) = 1/4
Find the domain of y considered as a function over the reals. (Enter your answer using interval notation.)
Find all the solutions, including imaginary solutions if they exist, of the following equation. x^4+17x^2+16=0
The solution to the equation is i , -i , 4i , -4i
What is Quadratic Equation?A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.
The roots of the quadratic equations are
x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
where ( b² - 4ac ) is the discriminant
when ( b² - 4ac ) is positive, we get two real solutions
when discriminant is zero we get just one real solution (both answers are the same)
when discriminant is negative we get a pair of complex solutions
Given data ,
Let the equation be represented as A
Now , the value of A is
x⁴ + 17x² + 16 = 0 be equation (1)
On simplifying the equation , we get
Substitute the value of x² = u
u² + 17u + 16 = 0
The roots of the quadratic equations are
u = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
u = [ -17 ± √ ( 289 - 4 ( 1 ) ( 16 ) ) ] / 2
On further simplification , we get
u = [ -17 ± √ ( 289 - 64 ) ] / 2
u = ( -17 ± √ 225 ) / 2
u = ( -17 ± 15 ) / 2
The two values of u are
u = ( -17 - 15 ) / 2
u = -16
And , u = ( -17 + 15 ) / 2
u = -1
Substitute the value of x² = u , we get
x² = -16
Taking square roots on both sides of the equation , we get
x = ± √ ( -16 )
x = ± 4i
And ,
x² = -1
x = ± i
So , the values of x are i , -i , -4i and 4i
Hence , the equations are solved
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We found the correlation coefficient for the comparison between height and weight in babies to be -0.12. That means
correlation is strong
the taller the baby, the heavier the baby
the taller the baby, the lighter the baby
Height and weight of babies are independent
The Correlation Coefficient -0.12 shows that The taller the baby, the lighter the baby.
What is Correlation Coefficient?A statistical measure called correlation shows how much two or more variables fluctuate in connection to one another.
A correlation coefficient is a statistical indicator of how well changes in one variable's value predict changes in another. When two variables are positively linked, the value either rises or falls together.
Given:
Correlation Coefficient = -0.12
As, The range of a correlation coefficient ranges from -1 to 1.
When the number is closer to -1 or closer to 1, there is a substantial association, and when the number is closer to 0, there is no link.
The correlation coefficient in this instance is -0.12, indicating a marginally negative correlation. The negative sign indicates that when one component changes, the other decreases, making a baby that is taller and lighter.
Hence, The taller the baby, the lighter the baby.
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Question 3 Multiple Choice Worth 2 points)
(04.05 MC)
B
A random sample of 75 juniors are asked whether they plan to attend Homecoming. Of these, 62 juniors said they will attend. What is the margin of error at 90% confidence and
its interpretation?
0.072; 90% of the time we will capture the true proportion of juniors who will attend Homecoming within 0.072 of the population proportion
0.072; we are 90% confident that the true proportion of juniors who will attend Homecoming is within 0.072 of the sample proportion
0.086; we are 90% confident that the true proportion of juniors who will attend Homecoming is within 0.086 of the sample proportion
0.086; 90% of the time we will capture the true proportion of juniors who will attend Homecoming within 0.086 of the population proportion
The margin of error at a 90% confidence level for this sample is approximately 0.086.
What is meant by margin of error?
In statistics, the margin of error is the level of inaccuracy in the findings of surveys using random samples.A wider margin of error in statistics denotes a lower chance of relying on a survey's or poll's findings, i.e., a lower level of confidence in the findings' ability to accurately reflect a population.When 62 out of 75 juniors in a random sample were asked if they would be attending homecoming, the margin of error at 90% confidence was roughly 0.086.This suggests that the true proportion of juniors who will attend homecoming is within 0.086 of the sample proportion, with a 90% confidence level.The standard error of the sample proportion is calculated using the following formula:
Standard error = √((sample proportion * (1 - sample proportion)) / sample size)
Plugging in the values for the sample proportion (62/75) and sample size (75), you get a standard error of approximately 0.072.
Multiplying the standard error by the critical value gives the margin of error:
The margin of error = critical value * standard error
= 1.645 * 0.072
= 0.086
Hence, the margin of error at a 90% confidence level for this sample is approximately 0.086.
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are the coordinates of the point on the directed line segment from (-3, -10) to (9, 5) that partitions the segment into a ratio of 1 to 2?
Answer:
An agency that occurs when a principal and an agent categorically agree to enter into an agency agreement with each other is known as which of the following?
help someone please!!!!!
62.8363 cm is the total circumference of the given circle.
What is Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a correct fraction is smaller than the denominator.
Given a circle that has an arc of 165 degrees and also given that arc has a circumference of 28.8cm
Circumference of 165 Degrees arc = 28.8cm
Circumference of 1 Degree arc = 28.8/165
Circumference of 1 Degree arc =0.17454
Since the total angle in a circle is 360 degrees
Thus from the fraction,
Circumference of 360 Degrees arc = 0.17454*360
Circumference of 365 Degrees arc = 62.8363 cm
Therefore, the total circumference of the given circle is 62.8363 cm.
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Suppose we want to choose letters, without replacement, from distinct letters. (a) If the order of the choices does not matter, how many ways can this be done? (b) If the order of the choices matters, how many ways can this be done?
The number of ways to choose 6 letters from a set of 13 is given as follows:
a) Order does not matter: 1716.
b) Order matters: 1,235,520.
When to use permutation and when to use combination?A combination is used when the order does not matter, hence, for item a, the number of ways is obtained as follows:
C(13,6) = 13!/(6!7!) = 1716.
A permutation is used when the order matters, hence, for item b, the number of ways is obtained as follows:
P(13,6) = 13!/7! = 1,235,520.
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Which of the following situations satisfies all the conditions of a binomial setting? (These conditions are: we know the number of repetitions, the outcome of each trial can be considered either a success or a failure, we know the probability of success or failure of any trial, and the probability doesn't change from trial to trial.)
A binomial setting is one in which there are a fixed number of trials, the outcome in each trial is either a success or a failure, the probability of success or failure is known, and the probability doesn't change from trial to trial.
A binomial setting is one in which we know the number of repetitions (n=20), the outcome of each trial can be considered either a success or a failure (p=0.4, q=1-p=0.6), and the probability of success or failure of any trial does not change from trial to trial (p=0.4, q=1-p=0.6).
For example, if we toss a coin 20 times, and we know the probability of success (head) is 0.4 and the probability of failure (tail) is 0.6, then this is a binomial setting. We can calculate the probability of getting exactly 10 heads in 20 tosses using the binomial formula: P(X=10) = (20C10) * (0.4^10) * (0.6^10) = 0.1775.
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Which equation matches the hanger diagram 2x=5
The equation that matches the hanger diagram is 2x=5
How to determine the equation of the diagramFrom the question, we have the following parameters that can be used in our computation:
The hanger diagram added as an attachment
From the diagram, we hav
Left = x + x
Right = 1 + 1 + 1 + 1 + 1
This gives
Left = 2x
Right = 5
So, we have
2x = 5
hence, the equation is 2x = 5
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What’s 23,760ft in miles
Answer:
So, there are 5,280 feet in a mile, so you will want to find out how many times 5, 280 will go into 23,760. To do this, you will divide.
23,760/5,280 = 4.5
There are 4.5 or 4 1/2 miles in 23,760 feet!
Step-by-step explanation:
Hope it helps! =D
You get a personal loan of $5,000 with 12% simple interest to be paid over 30 months. What is your monthly payment?
Work Shown:
A = P*(1+r*t)
A = 5000*(1+0.12*30/12)
A = 6500
The amount $6500 is what you pay back
Divided over 30 months gives 6500/30 = 216.67 dollars per month.
Side note: The value t is in years, so 30 months = 30/12 years.
A farmer is building a cylinder-shaped grain bin. The bin will be 10 feet tall and have a diameter of 12 feet. What is the surface area of the grain
bin in square feet? Use x = 3. 14. Round to the nearest hundredth if necessary.
Answer: 602.88 square feet
Equation for surface area of cylinder:
A = 2πrh+2πr^2
r = 6, because radius is half of the diameter
h = 10
A = 2(3.14)(6)(10) + 2(3.14)(6)^2
A = 376.8 + 226.08
A = 602.88 square feet
fill in the blank with a whole number with how many significant figures are in the following: a) 0.04550 b) 5.0003 c) 1.000 x 105 d) 0.00002 e) 1.01 x 10-2 f) 7
a) 0.04550 has four significant figures.
b) 5.0003 has four significant figures.
c) 1.000 x [tex]10^5[/tex] has four significant figures.
d) 0.00002 has one significant figure.
e) 1.01 x [tex]10^{-2}[/tex] has three significant figures.
f) 7 has one significant figure.
Significant figures are the digits in a number that are known with certainty, including one uncertain digit. In general, any non-zero digit is considered a significant figure, and zeros between non-zero digits are also significant figures.
Leading zeros before the first non-zero digit and trailing zeros after the last non-zero digit are not considered significant figures.
For example, in the given number 5.006, there is four significant number,
since all zeros between integers are always significant.
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A figure is made up of an equilateral triangle and a square of side 7cm. The perimeter of the figure is in stepes
12
75°
b
a
35°
Find all the unknow measures using the law of sines
Please help for math thank you!!!!!!!!
Answer:hie
hie
Step-by-step explanation:hie
Which set of ordered pairs is made up of points on the graph of the function below?
A.
(10,-1), (8,0), (1,6), (2,4)
B.
(10,-1), (8,0), (6,1), (4,2)
C.
(-1,10), (0,8), (1,6), (2,4)
D.
(-1,10), (8,0), (1,6), (4,2)
y=-2x+8
These points (-1,-8), (0,-6), (1,-4), and (2,-2) are the set of ordered pairs made up of points on the graph of the function y = 2x – 6.
What is the linear system?
A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.
The linear function is given below.
y = 2x – 6
The points (-1,-8), (0,-6), (1,-4), and (2,-2) lies on the equation.
As when the put the given points in the equation , than it satisfies the given condition.
Hence, these points (-1,-8), (0,-6), (1,-4), and (2,-2) are the set of ordered pairs made up of points on the graph of the function y = 2x – 6.
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Sorry you can’t see the photo clearly, I know it’s a reduction but what is the scale factor?
Answer:
1/2
Step-by-step explanation:
Triangle XYZ has endpoints (0,0) (0,8) and (4,6)
Triangle X'Y'Z' has endpoints (0,0) (0,4) and (2,3)
As you can see, all the coordinates from Triangle XYZ are halved to get the coordinates of Triangle X'Y'Z', so the scale factor is 1/2.
Type the missing numbers.
The missing numbers are -80 and -20.
What is Number Line?A number line is a straight line drawn horizontally where the numbers are placed at an equal intervals.
There is a number line in the question.
In a number line, numbers are arranged in equal intervals.
The constant interval here = -40 - -60 = 20
x - -100 = 20
x + 100 = 20
x = 20 - 100
x = -80
First missing number is -80.
y - -40 = 20
y + 40 = 20
y = 20 - 40
y = -20
Hence the numbers missing are -80 and -20.
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y÷15= 6 R 8
314÷y= 18 R8
What is y how can I solve this on my own for other problems?
Step-by-step explanation:
y÷15 or y/15 = 6
let's ignore the remainder for a moment.
y/15 = 6
y = 6×15 = 90
now, what does "remainder" mean ?
it means after a clean division there is a result, and there is still this additional remainder left.
the actual result of
a/b is
a/b = c + remainder/b
where c is an integer, and remainder is also an integer.
so, a = b×c + b×remainder/b = b×c + remainder
therefore,
y = 6×15 + 8 = 98
to check
98/15 = 6 remainder 8
so, it is correct
314÷y = 18 remainder 8
314 = 18y + 8
306 = 18y
y = 306/18 = 17
check
314 / 17 = 18 remainder 8
so, it is correct.
A study found that a student's GPA, g, is related to the number of hours worked each week, h, by the equation g=−0.0007h2+0.018h+3.02. Estimate the number of hours worked each week for a student with a GPA of 2.72.
After solving the given equation we know that students worked for 17 hours a week.
What are equations?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
So, the given equation is:
g=−0.0007h+0.018h+3.02
Now, we know that g is 2.72, so we can calculate for h as follows:
2.72=−0.0007h+0.018h+3.02
2.72 = -0.0173h + 3.02
- 0.3 = -0.0173h
h = -0.3/-0.0173
h = 17.34
Rounding off: 17 hours
Therefore, after solving the given equation we know that students worked for 17 hours a week.
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Data set: 8, 7, 10, 10, 17, 13, 7, 10
How much does the data set above vary?
Standard Deviation, σ: 3.152
Count, N: 8Sum, Σx: 82Mean, μ: 10.25Variance, σ2: 9.9375What a standard deviation means?The term "standard deviation" (or "σ") refers to a measurement of the data's dispersion from the mean. A low standard deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.
Given, Data set: 8, 7, 10, 10, 17, 13, 7, 10
variance:
σ² = Σ(xi - μ)²/N
σ²= ((8 - 10.25)² + … + (10 - 10.25)²)/8
σ²= 79.5/8
σ² =9.9375
σ = √9.9375
σ = 3.1523
The sampling mean most likely follows a normal distribution. In this case, the standard error of the mean can be calculated using the following equation:
σ = σ/√N
σ = 1.114
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Nicole gets paid $2,900 per month. She pays $638 a month for rent. What percent of her monthly pay goes to rent?
21.931% of Nicole's monthly pay goes to rent.
percent = (638/2900) x 100
percent = 0.21931 x 100
percent = 21.931%
find the center of mass (relative to (0,0)) for the spheres with the following masses and locations: m1
The center of mass is (2,1) since m1 has a mass of 10 and is located at (4,2). The center of mass is the average of the locations of the masses, weighted by the masses.
We can calculate the center of mass of these two spheres by using the formula c = (m1*L1 + m2*L2) / (m1 + m2). m1 is the mass of the first sphere, and L1 is the location of the first sphere. In this case, m1 is 10, and L1 is (4,2). We assume that m2 is 0 and L2 is (0,0). Plugging these values into the formula, we get c = (10*(4,2) + 0*(0,0)) / (10 + 0), which is (2,1). This is because the center of mass is the average of the locations of the masses, weighted by the masses. In this case, the only mass is 10, and it is located at (4,2). Therefore, the center of mass is simply (4,2) since it is the only mass that contributes to the calculation.
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The complete question: find the center of mass (relative to (0,0)) for the spheres with the following masses and locations: m1 = 10, L1 = ( 4,2)
The period (T) of a pendulum is related to the length (L) of the pendulum and acceleration due to gravity (g) by the formula T=2π√L/g. If gravity is 32 ft/s2 and the period is 1 second, find the approximate length of the pendulum. Round to the nearest inch. Note: 12 in. = 1 ft.
The approximate length of the pendulum is 2 feet.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given that period (T) of a pendulum is related to the length (L) of the pendulum and
acceleration due to gravity (g) by the formula T=2π√L/g.
g=32 ft/s2
t=1 sec
We need to find L
1=2×3.14√L/32
32=6.28√L
32/6.28=√L
5.09=√L
Square on both sides
L=25.9 inches
L=2 feet because 12 inches = 1ft.
Hence, the approximate length of the pendulum is 2 feet.
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6. To play a board game, there must be at least 4 people on each team. You
divide your friends into 3 groups. Write and solve an inequality to represent
the number of friends playing the game.
The inequality that best represent the expression is x ≥ 12
What is inequality?A relationship between two expressions or values that are not equal to each other is called 'inequality. ' So, a lack of balance results in inequality.
The sign for greater than is >
The sign for less than <
The sign for greater than or equal to is ≥
The sign for less than or equal to is ≤
There must be at least 4 people in the group. This means that a group member must greater or equal to 4. There are 3 groups.
Represent the value of the number of friends by x
then, x/3 ≥ 4
therefore x ≥ 12
this means that the number of friends must be greater or equal to 12
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which of these has a slope of -3
Answer:
Option 3. Look at my screenshot so you can see the circled one
Step-by-step explanation:
The reasoning behind this is slope is rise over run (rise/run)
Graph 3 goes down 3 so -3 and runs 1 over so (-3/1)
This is simplified as -3.
I hope this helps you out :D
Distribute to create an equivalent expression with the fewest symbols possible.
\dfrac12(2a - 6b+ 8) =
2
1
(2a−6b+8
The equivalent expression for 1/2(2a - 6b+ 8) will be a - 3b + 4
How to calculate the equivalent expression?Expressions that are equivalent do the same thing even though they have different appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).
When two expressions can be reduced to a single third expression or when one of the expressions can be written in the same way as the other, they are said to be equivalent. When values are substituted for the variable and both expressions yield the same result, you can also tell if two expressions are equivalent.
This will be:
1/2(2a - 6b+ 8)
= a - 3b + 4
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Complete question
Distribute to create an equivalent expression with the fewest symbols possible 1/2(2a - 6b+ 8) =
In 2012, the population of a city was 5.89 million. The exponential growth rate was 3.42% per year.
a) Find the exponential growth function.
b) Estimate the population of the city in 2018.
c) When will the population of the city be 8 million?
d) Find the doubling time.
The general form of exponential formula is y = a * b^x, the population of the city in 2018 6.058454 million.(c) the population of the city be 8 million19.38548963 years.
How to find the population?In this problem, a = 5.21 million and b = 1 + 2.86%/100 = 1.0286.
a is a constant.b is the basex is the exponent.In 2012, the population was 5.89 million.
In 2018, the population will be 5.89 * 1.0286 ^ (2018 - 2012) = 5.89 * 1.0286 ^ 6 = 6.058454 million.
To find when the population will be 9 million, your equation becomes:
9 = 5.21 * 1.0286 ^ x divide both sides of this formula by 5.21 to get:
9/5.21 = 1.0286 ^ x take the log of both sides of the equation to get:
log(9/5.21) = log(1.0286^x) since log(1.0286^x) = x * log(1.0286), the equation becomes:
log(9/5.21) = x * log(1.0286)
Divide both sides of this equation by log(1.0286) to get:
log(9/5.21) / log(1.0286) = x
Solve for x to get:
x = 19.38548963.
Confirm by replace x in the original equation to get:
9 = 5.21 * 1.0286 ^ 19.38548963 becomes 9 = 9.
This confirms the value of x is good.
The population will grow to 9 million in 19.38548963 years.
To find when the population will double, the equation becomes:
2 = 1 * 1.0286 ^ x
Simplify to get:
2 = 1.0286 ^ x
Take the log of both sides of this equation to get:
log(2) = log(1.0286 ^ x)
Since log(1.0286 ^ x) = x * log(1.0286), this equation becomes:
log(2) = x * log(1.0286)
solve for x to get:
x = log(2) / log(1.0286) = 24.5808602.
to confirm this is true, replace x in the original equation to get:
2 = 1.0286 ^ 24.5808602 which becomes 2 = 2.
This confirms the value of x is true.
The population will double in 24.5808602 years.
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